Term Rewriting System R:
[x]
f(h(x)) -> f(i(x))
g(i(x)) -> g(h(x))
h(a) -> b
i(a) -> b
Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(h(x)) -> F(i(x))
F(h(x)) -> I(x)
G(i(x)) -> G(h(x))
G(i(x)) -> H(x)
Furthermore, R contains two SCCs.
R
↳DPs
→DP Problem 1
↳Argument Filtering and Ordering
→DP Problem 2
↳AFS
Dependency Pair:
F(h(x)) -> F(i(x))
Rules:
f(h(x)) -> f(i(x))
g(i(x)) -> g(h(x))
h(a) -> b
i(a) -> b
The following dependency pair can be strictly oriented:
F(h(x)) -> F(i(x))
The following usable rule w.r.t. to the AFS can be oriented:
i(a) -> b
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
h > {i, b}
resulting in one new DP problem.
Used Argument Filtering System: F(x1) -> F(x1)
h(x1) -> h(x1)
i(x1) -> i(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 3
↳Dependency Graph
→DP Problem 2
↳AFS
Dependency Pair:
Rules:
f(h(x)) -> f(i(x))
g(i(x)) -> g(h(x))
h(a) -> b
i(a) -> b
Using the Dependency Graph resulted in no new DP problems.
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Argument Filtering and Ordering
Dependency Pair:
G(i(x)) -> G(h(x))
Rules:
f(h(x)) -> f(i(x))
g(i(x)) -> g(h(x))
h(a) -> b
i(a) -> b
The following dependency pair can be strictly oriented:
G(i(x)) -> G(h(x))
The following usable rule w.r.t. to the AFS can be oriented:
h(a) -> b
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
i > h
{a, b}
resulting in one new DP problem.
Used Argument Filtering System: G(x1) -> G(x1)
i(x1) -> i(x1)
h(x1) -> h(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 4
↳Dependency Graph
Dependency Pair:
Rules:
f(h(x)) -> f(i(x))
g(i(x)) -> g(h(x))
h(a) -> b
i(a) -> b
Using the Dependency Graph resulted in no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes